Data.Approximate.Numerics:blog from approximate-0.2.2.1

Percentage Accurate: 99.7% → 99.9%

Time: 10.0s

Alternatives: 14

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))

C

double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * (x - 1.0d0)) / ((x + 1.0d0) + (4.0d0 * sqrt(x)))
end function

Java

public static double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * Math.sqrt(x)));
}

Python

def code(x):
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * math.sqrt(x)))

Julia

function code(x)
	return Float64(Float64(6.0 * Float64(x - 1.0)) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
end

Wolfram

code[x_] := N[(N[(6.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (+ x -1.0) (+ x (fma 4.0 (sqrt x) 1.0))) 6.0))

C

double code(double x) {
	return ((x + -1.0) / (x + fma(4.0, sqrt(x), 1.0))) * 6.0;
}

Julia

function code(x)
	return Float64(Float64(Float64(x + -1.0) / Float64(x + fma(4.0, sqrt(x), 1.0))) * 6.0)
end

Wolfram

code[x_] := N[(N[(N[(x + -1.0), $MachinePrecision] / N[(x + N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{6 \cdot \frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6} \]

   6. lower-/.f64 100.0

      \[\leadsto \color{blue}{\frac{x - 1}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot 6 \]

   7. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{x - 1}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6 \]

   8. sub-neg N/A

      \[\leadsto \frac{\color{blue}{x + \left(\mathsf{neg}\left(1\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6 \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{x + \left(\mathsf{neg}\left(1\right)\right)}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6 \]

   10. metadata-eval 100.0

       \[\leadsto \frac{x + \color{blue}{-1}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \cdot 6 \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{x + -1}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \cdot 6 \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{x + -1}{\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}} \cdot 6 \]

   13. associate-+l+ N/A

       \[\leadsto \frac{x + -1}{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}} \cdot 6 \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{x + -1}{\color{blue}{x + \left(1 + 4 \cdot \sqrt{x}\right)}} \cdot 6 \]

   15. +-commutative N/A

       \[\leadsto \frac{x + -1}{x + \color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}} \cdot 6 \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{x + -1}{x + \left(\color{blue}{4 \cdot \sqrt{x}} + 1\right)} \cdot 6 \]

   17. lower-fma.f64 100.0

       \[\leadsto \frac{x + -1}{x + \color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \cdot 6 \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{x + -1}{x + \mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq 2:\\ \;\;\;\;6 \cdot \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{1 + \mathsf{fma}\left(4, \sqrt{x}, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (+ x -1.0) 6.0) (+ (+ x 1.0) (* 4.0 (sqrt x)))) 2.0)
   (* 6.0 (/ (+ x -1.0) (fma 4.0 (sqrt x) 1.0)))
   (/ (* x 6.0) (+ 1.0 (fma 4.0 (sqrt x) x)))))

C

double code(double x) {
	double tmp;
	if ((((x + -1.0) * 6.0) / ((x + 1.0) + (4.0 * sqrt(x)))) <= 2.0) {
		tmp = 6.0 * ((x + -1.0) / fma(4.0, sqrt(x), 1.0));
	} else {
		tmp = (x * 6.0) / (1.0 + fma(4.0, sqrt(x), x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x + -1.0) * 6.0) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= 2.0)
		tmp = Float64(6.0 * Float64(Float64(x + -1.0) / fma(4.0, sqrt(x), 1.0)));
	else
		tmp = Float64(Float64(x * 6.0) / Float64(1.0 + fma(4.0, sqrt(x), x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < 2

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

      3. lower-sqrt.f64 98.4

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \color{blue}{\sqrt{x}}, 1\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \color{blue}{\left(x + -1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{6 \cdot \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6} \]

      10. lower-/.f64 98.4

          \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \cdot 6 \]

   7. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6} \]

   if 2 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{4 \cdot \sqrt{x} + \color{blue}{\left(x + 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(\color{blue}{4 \cdot \sqrt{x}} + x\right) + 1} \]

      7. lower-fma.f64 99.7

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)} + 1} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{6 \cdot x}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

      2. lower-*.f64 98.7

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   7. Applied rewrites 98.7%

      \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq 2:\\ \;\;\;\;6 \cdot \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{1 + \mathsf{fma}\left(4, \sqrt{x}, x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -5:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (+ x -1.0) 6.0) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -5.0)
   (/ -6.0 (fma (sqrt x) 4.0 (+ x 1.0)))
   (/ (* x 6.0) (fma 4.0 (sqrt x) 1.0))))

C

double code(double x) {
	double tmp;
	if ((((x + -1.0) * 6.0) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -5.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (x + 1.0));
	} else {
		tmp = (x * 6.0) / fma(4.0, sqrt(x), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x + -1.0) * 6.0) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -5.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x * 6.0) / fma(4.0, sqrt(x), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -5

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x}} + \left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{\sqrt{x} \cdot 4} + \left(x + 1\right)} \]

      5. lower-fma.f64 99.0

         \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

   if -5 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

      3. lower-sqrt.f64 7.1

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \color{blue}{\sqrt{x}}, 1\right)} \]

   5. Applied rewrites 7.1%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{6 \cdot x}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

      2. lower-*.f64 7.1

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   8. Applied rewrites 7.1%

      \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -5:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \sqrt{x}\\ \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + t\_0} \leq -5:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 4.0 (sqrt x))))
   (if (<= (/ (* (+ x -1.0) 6.0) (+ (+ x 1.0) t_0)) -5.0)
     (/ -6.0 (fma (sqrt x) 4.0 (+ x 1.0)))
     (/ (* x 6.0) t_0))))

C

double code(double x) {
	double t_0 = 4.0 * sqrt(x);
	double tmp;
	if ((((x + -1.0) * 6.0) / ((x + 1.0) + t_0)) <= -5.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (x + 1.0));
	} else {
		tmp = (x * 6.0) / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(4.0 * sqrt(x))
	tmp = 0.0
	if (Float64(Float64(Float64(x + -1.0) * 6.0) / Float64(Float64(x + 1.0) + t_0)) <= -5.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x * 6.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -5.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 6.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -5

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x}} + \left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{\sqrt{x} \cdot 4} + \left(x + 1\right)} \]

      5. lower-fma.f64 99.0

         \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

   if -5 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

      3. lower-sqrt.f64 7.1

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \color{blue}{\sqrt{x}}, 1\right)} \]

   5. Applied rewrites 7.1%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{6 \cdot x}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

      2. lower-*.f64 7.1

         \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   8. Applied rewrites 7.1%

      \[\leadsto \frac{\color{blue}{x \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot 6}{4 \cdot \color{blue}{\sqrt{x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.1%

       \[\leadsto \frac{x \cdot 6}{\sqrt{x} \cdot \color{blue}{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -5:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 6}{4 \cdot \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -5:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (* (+ x -1.0) 6.0) (+ (+ x 1.0) (* 4.0 (sqrt x)))) -5.0)
   (/ 6.0 (fma (sqrt x) -4.0 -1.0))
   (/ (fma (sqrt x) 1.5 0.375) x)))

C

double code(double x) {
	double tmp;
	if ((((x + -1.0) * 6.0) / ((x + 1.0) + (4.0 * sqrt(x)))) <= -5.0) {
		tmp = 6.0 / fma(sqrt(x), -4.0, -1.0);
	} else {
		tmp = fma(sqrt(x), 1.5, 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(Float64(x + -1.0) * 6.0) / Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x)))) <= -5.0)
		tmp = Float64(6.0 / fma(sqrt(x), -4.0, -1.0));
	else
		tmp = Float64(fma(sqrt(x), 1.5, 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[(x + -1.0), $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5.0], N[(6.0 / N[(N[Sqrt[x], $MachinePrecision] * -4.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 1.5 + 0.375), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x)))) < -5

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 98.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   if -5 < (/.f64 (*.f64 #s(literal 6 binary64) (-.f64 x #s(literal 1 binary64))) (+.f64 (+.f64 x #s(literal 1 binary64)) (*.f64 #s(literal 4 binary64) (sqrt.f64 x))))

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 1.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.9%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-1.5} \]

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-3}{2} \cdot \sqrt{x} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x + -1\right) \cdot 6}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \leq -5:\\ \;\;\;\;\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (fma (sqrt x) 4.0 (+ x 1.0)))
   (/ (fma (sqrt x) 1.5 0.375) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / fma(sqrt(x), 4.0, (x + 1.0));
	} else {
		tmp = fma(sqrt(x), 1.5, 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / fma(sqrt(x), 4.0, Float64(x + 1.0)));
	else
		tmp = Float64(fma(sqrt(x), 1.5, 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(N[Sqrt[x], $MachinePrecision] * 4.0 + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 1.5 + 0.375), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{-6}}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-6}{\color{blue}{4 \cdot \sqrt{x}} + \left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{-6}{\color{blue}{\sqrt{x} \cdot 4} + \left(x + 1\right)} \]

      5. lower-fma.f64 99.0

         \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{-6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 1.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.9%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-1.5} \]

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-3}{2} \cdot \sqrt{x} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + \mathsf{fma}\left(4, \sqrt{x}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (/ -6.0 (+ 1.0 (fma 4.0 (sqrt x) x)))
   (/ (fma (sqrt x) 1.5 0.375) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -6.0 / (1.0 + fma(4.0, sqrt(x), x));
	} else {
		tmp = fma(sqrt(x), 1.5, 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-6.0 / Float64(1.0 + fma(4.0, sqrt(x), x)));
	else
		tmp = Float64(fma(sqrt(x), 1.5, 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-6.0 / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 1.5 + 0.375), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{4 \cdot \sqrt{x} + \color{blue}{\left(x + 1\right)}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(\color{blue}{4 \cdot \sqrt{x}} + x\right) + 1} \]

      7. lower-fma.f64 100.0

         \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)} + 1} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{-6}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 1.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.9%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-1.5} \]

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-3}{2} \cdot \sqrt{x} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-6}{1 + \mathsf{fma}\left(4, \sqrt{x}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, 6, -6\right)}{1 + \mathsf{fma}\left(4, \sqrt{x}, x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma x 6.0 -6.0) (+ 1.0 (fma 4.0 (sqrt x) x))))

C

double code(double x) {
	return fma(x, 6.0, -6.0) / (1.0 + fma(4.0, sqrt(x), x));
}

Julia

function code(x)
	return Float64(fma(x, 6.0, -6.0) / Float64(1.0 + fma(4.0, sqrt(x), x)))
end

Wolfram

code[x_] := N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(1.0 + N[(4.0 * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + \left(x + 1\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{4 \cdot \sqrt{x} + \color{blue}{\left(x + 1\right)}} \]

   4. associate-+r+ N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\left(4 \cdot \sqrt{x} + x\right) + 1}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\left(\color{blue}{4 \cdot \sqrt{x}} + x\right) + 1} \]

   7. lower-fma.f64 99.8

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right)} + 1} \]

4. Applied rewrites 99.8%

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   3. sub-neg N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   4. metadata-eval N/A

      \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{x \cdot 6 + -1 \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{x \cdot 6 + \color{blue}{-6}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   7. metadata-eval N/A

      \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(6\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

   9. metadata-eval 99.8

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

6. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, -6\right)}}{\mathsf{fma}\left(4, \sqrt{x}, x\right) + 1} \]

7. Final simplification 99.8%

   \[\leadsto \frac{\mathsf{fma}\left(x, 6, -6\right)}{1 + \mathsf{fma}\left(4, \sqrt{x}, x\right)} \]
8. Add Preprocessing

Alternative 9: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, -6, 6\right)}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right) - x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma x -6.0 6.0) (- (fma (sqrt x) -4.0 -1.0) x)))

C

double code(double x) {
	return fma(x, -6.0, 6.0) / (fma(sqrt(x), -4.0, -1.0) - x);
}

Julia

function code(x)
	return Float64(fma(x, -6.0, 6.0) / Float64(fma(sqrt(x), -4.0, -1.0) - x))
end

Wolfram

code[x_] := N[(N[(x * -6.0 + 6.0), $MachinePrecision] / N[(N[(N[Sqrt[x], $MachinePrecision] * -4.0 + -1.0), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(6 \cdot \left(x - 1\right)\right)}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(6 \cdot \left(x - 1\right)\right)}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{6 \cdot \left(x - 1\right)}\right)}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(6\right)\right) \cdot \left(x - 1\right)}}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(6\right)\right) \cdot \color{blue}{\left(x - 1\right)}}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(6\right)\right) \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   8. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(6\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(6\right)\right)}}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{x \cdot \left(\mathsf{neg}\left(6\right)\right) + \color{blue}{-1} \cdot \left(\mathsf{neg}\left(6\right)\right)}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{x \cdot \left(\mathsf{neg}\left(6\right)\right) + -1 \cdot \color{blue}{-6}}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{x \cdot \left(\mathsf{neg}\left(6\right)\right) + \color{blue}{6}}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(6\right), 6\right)}}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-6}, 6\right)}{\mathsf{neg}\left(\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)\right)} \]

   14. neg-sub0 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{\color{blue}{0 - \left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{0 - \color{blue}{\left(\left(x + 1\right) + 4 \cdot \sqrt{x}\right)}} \]

   16. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{0 - \left(\color{blue}{\left(x + 1\right)} + 4 \cdot \sqrt{x}\right)} \]

   17. associate-+l+ N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{0 - \color{blue}{\left(x + \left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

   18. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{0 - \color{blue}{\left(\left(1 + 4 \cdot \sqrt{x}\right) + x\right)}} \]

   19. associate--r+ N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{\color{blue}{\left(0 - \left(1 + 4 \cdot \sqrt{x}\right)\right) - x}} \]

   20. neg-sub0 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right)} - x} \]

   21. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, -6, 6\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)\right) - x}} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, -6, 6\right)}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right) - x}} \]
5. Add Preprocessing

Alternative 10: 52.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 6 \cdot \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 6.0 (/ (+ x -1.0) (fma 4.0 (sqrt x) 1.0))))

C

double code(double x) {
	return 6.0 * ((x + -1.0) / fma(4.0, sqrt(x), 1.0));
}

Julia

function code(x)
	return Float64(6.0 * Float64(Float64(x + -1.0) / fma(4.0, sqrt(x), 1.0)))
end

Wolfram

code[x_] := N[(6.0 * N[(N[(x + -1.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

   3. lower-sqrt.f64 53.0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \color{blue}{\sqrt{x}}, 1\right)} \]

5. Applied rewrites 53.0%

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   4. sub-neg N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x + -1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{6 \cdot \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6} \]

   10. lower-/.f64 53.0

       \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \cdot 6 \]

7. Applied rewrites 53.0%

   \[\leadsto \color{blue}{\frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \cdot 6} \]

8. Final simplification 53.0%

   \[\leadsto 6 \cdot \frac{x + -1}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]
9. Add Preprocessing

Alternative 11: 7.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -1.5 (sqrt x)) (/ (fma (sqrt x) 1.5 0.375) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = fma(sqrt(x), 1.5, 0.375) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-1.5 / sqrt(x));
	else
		tmp = Float64(fma(sqrt(x), 1.5, 0.375) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 1.5 + 0.375), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 98.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 6.8%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-1.5} \]
   9. Step-by-step derivation

   10. Applied rewrites 6.8%

       \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 1.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.9%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-1.5} \]

   9. Taylor expanded in x around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-3}{2} \cdot \sqrt{x} + \frac{3}{8} \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}}}{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 7.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, 1.5, 0.375\right)}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 52.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, 6, -6\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x 6.0 -6.0) (fma 4.0 (sqrt x) 1.0)))

C

double code(double x) {
	return fma(x, 6.0, -6.0) / fma(4.0, sqrt(x), 1.0);
}

Julia

function code(x)
	return Float64(fma(x, 6.0, -6.0) / fma(4.0, sqrt(x), 1.0))
end

Wolfram

code[x_] := N[(N[(x * 6.0 + -6.0), $MachinePrecision] / N[(4.0 * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{1 + 4 \cdot \sqrt{x}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{4 \cdot \sqrt{x} + 1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]

   3. lower-sqrt.f64 53.0

      \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\mathsf{fma}\left(4, \color{blue}{\sqrt{x}}, 1\right)} \]

5. Applied rewrites 53.0%

   \[\leadsto \frac{6 \cdot \left(x - 1\right)}{\color{blue}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{6 \cdot \left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x - 1\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   3. sub-neg N/A

      \[\leadsto \frac{6 \cdot \color{blue}{\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{6 \cdot \left(x + \color{blue}{-1}\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{x \cdot 6 + -1 \cdot 6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{x \cdot 6 + \color{blue}{-6}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{x \cdot 6 + \color{blue}{\left(\mathsf{neg}\left(6\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, \mathsf{neg}\left(6\right)\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

   9. metadata-eval 53.0

      \[\leadsto \frac{\mathsf{fma}\left(x, 6, \color{blue}{-6}\right)}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]

7. Applied rewrites 53.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 6, -6\right)}}{\mathsf{fma}\left(4, \sqrt{x}, 1\right)} \]
8. Add Preprocessing

Alternative 13: 7.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0) (/ -1.5 (sqrt x)) (* 1.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / sqrt(x);
	} else {
		tmp = 1.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (-1.5d0) / sqrt(x)
    else
        tmp = 1.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -1.5 / Math.sqrt(x);
	} else {
		tmp = 1.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -1.5 / math.sqrt(x)
	else:
		tmp = 1.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-1.5 / sqrt(x));
	else
		tmp = Float64(1.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -1.5 / sqrt(x);
	else
		tmp = 1.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 98.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 6.8%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-1.5} \]
   9. Step-by-step derivation

   10. Applied rewrites 6.8%

       \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]

   if 1 < x

   1. Initial program 99.7%

      \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

      14. metadata-eval 1.9

          \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

   5. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \frac{3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 7.0%

      \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-1.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 4.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1.5}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.5 (sqrt x)))

C

double code(double x) {
	return -1.5 / sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.5d0) / sqrt(x)
end function

Java

public static double code(double x) {
	return -1.5 / Math.sqrt(x);
}

Python

def code(x):
	return -1.5 / math.sqrt(x)

Julia

function code(x)
	return Float64(-1.5 / sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = -1.5 / sqrt(x);
end

Wolfram

code[x_] := N[(-1.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-6}{1 + 4 \cdot \sqrt{x}}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(6\right)}}{1 + 4 \cdot \sqrt{x}} \]

   2. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{6}{1 + 4 \cdot \sqrt{x}}\right)} \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{6}{\mathsf{neg}\left(\left(1 + 4 \cdot \sqrt{x}\right)\right)}} \]

   5. +-commutative N/A

      \[\leadsto \frac{6}{\mathsf{neg}\left(\color{blue}{\left(4 \cdot \sqrt{x} + 1\right)}\right)} \]

   6. distribute-neg-in N/A

      \[\leadsto \frac{6}{\color{blue}{\left(\mathsf{neg}\left(4 \cdot \sqrt{x}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{6}{\left(\mathsf{neg}\left(\color{blue}{\sqrt{x} \cdot 4}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \frac{6}{\color{blue}{\sqrt{x} \cdot \left(\mathsf{neg}\left(4\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{-4} + \left(\mathsf{neg}\left(1\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{6}{\sqrt{x} \cdot \color{blue}{\left(4 \cdot -1\right)} + \left(\mathsf{neg}\left(1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{6}{\sqrt{x} \cdot \left(4 \cdot -1\right) + \color{blue}{-1}} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{6}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4 \cdot -1, -1\right)}} \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \frac{6}{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, 4 \cdot -1, -1\right)} \]

   14. metadata-eval 50.4

       \[\leadsto \frac{6}{\mathsf{fma}\left(\sqrt{x}, \color{blue}{-4}, -1\right)} \]

5. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\frac{6}{\mathsf{fma}\left(\sqrt{x}, -4, -1\right)}} \]

6. Taylor expanded in x around inf

   \[\leadsto \frac{-3}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]
7. Step-by-step derivation

8. Applied rewrites 4.3%

   \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{-1.5} \]
9. Step-by-step derivation

10. Applied rewrites 4.3%

    \[\leadsto \color{blue}{\frac{-1.5}{\sqrt{x}}} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0))))

C

double code(double x) {
	return 6.0 / (((x + 1.0) + (4.0 * sqrt(x))) / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0 / (((x + 1.0d0) + (4.0d0 * sqrt(x))) / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return 6.0 / (((x + 1.0) + (4.0 * Math.sqrt(x))) / (x - 1.0));
}

Python

def code(x):
	return 6.0 / (((x + 1.0) + (4.0 * math.sqrt(x))) / (x - 1.0))

Julia

function code(x)
	return Float64(6.0 / Float64(Float64(Float64(x + 1.0) + Float64(4.0 * sqrt(x))) / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 6.0 / (((x + 1.0) + (4.0 * sqrt(x))) / (x - 1.0));
end

Wolfram

code[x_] := N[(6.0 / N[(N[(N[(x + 1.0), $MachinePrecision] + N[(4.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024238 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1))))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))